1. Angle-domain Wave-equation Reflection Traveltime Inversion
Sanzong Zhang, Yi Luo and Gerard Schuster
(1) KAUST, (2) Aramco 1 2
Good afternoon, everyone. The title of my talk is wave-equation reflection traveltime inversion. The coauthors are Dr. Yi Luo from Aramco, and Prof. Schuster.

2. Outline
Introduction
Theory and method
Numerical examples
Conclusions

3. Outline
Introduction
Theory and method
Numerical examples
Conclusions

4. Velocity Inversion Methods
Data space
Image space
Ray-based tomography
Full Waveform inversion
Ray-based MVA
Wave-equ. MVA
Inversion
Wave-equ. Reflection traveltime inversion
(Tomography)
Wave-equ. Reflection traveltime inversion
The current velocity inversion methods can be divided into two types. One type defines the objective function in data space, such as traveltime tomography and full waveform inversion. The other type defines the objective function in model space, such as ray-based MVA and wave-equation-based MVA.
(MVA)

5. Problem
The waveform (image) residual is highly nonlinear with respect to velocity change.
Model Parameter 𝜀 - 2
e =
Pred. data – Obs. data ∆𝜏
The traveltime misfit function enjoys a somewhat linear relationship with velocity change.

6. Angle-domain Wave-equation Reflection Traveltime Inversion
Traveltime inversion without high-frequency approximation
Misfit function somewhat linear with respect to velocity perturbation.
Wave-equation inversion less sensitive to amplitude
Multi-arrival traveltime inversion
Beam-based reflection traveltime inversion
To exploit the strengths and avoid the weaknesses of both traveltime tomography and full waveform inversion, wave-equation-based transimission traveltime inversion was developed to invert the velocity model. This method tried to invert the transmission traveltime informaiton using the method similar to full waveform inversion.

7. Outline
Introduction
Theory and method
Numerical examples
Conclusions

8. Wave-equation Transmission
Traveltime Inversion
1). Observed data 𝑝 𝑜𝑏𝑠 5
Time (s)
2). Calculated data
Time (s) 5
𝑝 𝑐𝑎𝑙𝑐
-1.5
1.5
Lag time (s)
3). 𝑝 𝑜𝑏𝑠
𝑝 𝑐𝑎𝑙𝑐 ∆𝜏
For transmission tomography, the receiver records the first arrival initiated by the source. The misfit function is defined to minimize the traveltime difference between the observed data and the calculated data. This misfit function is similar to that of ray-based traveltime tomography.
4). Smear time delay ∆𝜏
along wavepath

9. Angle-domain Wave-equation Reflection Traveltime Inversion
Suboffset-domain crosscorrelation function :
𝑓 𝑥,𝑧,ℎ,𝜏 = 𝑑 x 𝑠 𝑝 𝑓 (𝑥−ℎ,𝑧,𝑡+𝜏| x 𝑠 ) 𝑝 𝑏 (𝑥+ℎ,𝑧,𝑡| x 𝑠 )𝑑𝑡 g s x
x-h
x+h
𝑝 𝑓 (𝑥−ℎ,𝑧,𝑡| x 𝑠 )
𝑝 𝑏 (𝑥+ℎ,𝑧,𝑡| x 𝑠 )
𝑝 𝑓: 𝑓𝑜𝑤𝑎𝑟𝑑 𝑝𝑟𝑜𝑝𝑎𝑔𝑎𝑡𝑒𝑑 𝑑𝑎𝑡𝑎
𝑝 𝑏: 𝑏𝑎𝑐𝑘𝑤𝑎𝑟𝑑 𝑝𝑟𝑜𝑝𝑎𝑔𝑎𝑡𝑒𝑑 𝑑𝑎𝑡𝑎
ℎ: 𝑠𝑢𝑏𝑜𝑓𝑓𝑠𝑒𝑡
𝜏: time shift

10. Angle-domain Crosscorrelation
Angle-domain CIG decomposition (slant stack ):
𝑓 𝑥,𝑧,𝜃,𝜏 = 𝑓 𝑥,𝑧+ℎ tan 𝜃 ,ℎ,𝜏 𝐱 𝑠 𝑑ℎ
angle-domain
suboffset-domain
Angle-domain crosscorrelation function :
𝑓 𝑥,𝑧,𝜃,𝜏 = 𝑑 𝐱 𝑠 𝑑ℎ 𝑝 𝑓 𝑥−ℎ,𝑧+ℎ tan 𝜃 ,𝑡+𝜏 𝐱 𝑠
𝑝 𝑏 (𝑥+ℎ,𝑧+ℎ tan 𝜃 ,𝑡| 𝐱 𝑠 ) 𝑑𝑡

11. Angle-domain Crosscorrelation:
physical meaning
𝑓 𝑥,𝑧,𝜃,𝜏=0 = 𝑑 𝐱 𝑠 𝑑ℎ 𝑝 𝑓 𝑥−ℎ,𝑧+ℎ tan 𝜃 ,𝑡 𝐱 𝑠
𝑝 𝑏 (𝑥+ℎ,𝑧+ℎ tan 𝜃 ,𝑡| 𝐱 𝑠 ) 𝑑𝑡 𝑥 𝑧 𝜃
𝑝 𝑓 (𝑥−ℎ,𝑧+ℎ tan 𝜃 ,𝑡| 𝐱 𝑠 )
Local plane wave 𝜃 𝑥 𝑧
𝑝 𝑏 (𝑥+ℎ,𝑧+ℎ tan 𝜃 ,𝑡| 𝐱 𝑠 )
Local plane wave
Angle-domain crosscorrelation is the crosscorrelation
between downgoing and upgoing beams with a certain angle.
The time delay for multi-arrivals is available in angle
-domain crosscorrelation function .

12. Angle-domain Wave-equation Reflection Traveltime Inversion
Objective function:
𝜀= 𝐱 𝜽 ∆𝜏(𝐱,𝜃) 𝟐
Velocity update:
𝑐 𝑘+1 (x)= 𝑐 𝑘 (x) + 𝛼 𝑘 ∙ 𝛾 𝑘 (x)
Gradient function:
𝛾 𝑘 (x)= − 𝜕𝜀 𝜕𝑐(𝐱) =− 𝐱 𝜽 ∆𝜏 𝜕(∆𝜏) 𝜕𝑐(𝐱)
Traveltime wavepath

13. Traveltime Wavepath Angle-domain time delay ∆𝜏 𝜃
𝑓 𝑥,𝑧,𝜃,∆𝜏 = max −𝑇<𝜏<𝑇 𝑓 𝑥,𝑧,𝜃,𝜏
𝑓 𝑥,𝑧,𝜃,∆𝜏 = m𝑖𝑛 −𝑇<𝜏<𝑇 𝑓 𝑥,𝑧,𝜃,𝜏
Angle-domain connective function
𝑓 ∆𝜏 = 𝜕𝑓(𝑥,𝑧,𝜃,𝜏) 𝜕𝜏 𝜏=∆𝜏 =0
Traveltime wavepath
𝜕(∆𝜏) 𝜕𝑐(𝑥) =− 𝜕 𝑓 ∆𝜏 𝜕𝑐(𝑥) 𝜕 𝑓 ∆𝜏 𝜕(∆𝜏)

14. Transforming CSG Data  Xwell Trans. Data= +
reflection
transmission
Src-side Xwell Data
Redatuming data
source
Redatuming source
Observed data
Rec-side Xwell Data

15. Workflow
Forward propagate source to trial image points and get downgoing beams
Backward propagate observed reflection data from geophonses to trial image points , and get upgoing beams
Crosscorrelate downgoing beam and upgoing beam, and pick angle-domain time delay ∆𝝉 𝒛 𝜽
Smear time dealy along wavepath to update velocity model

16. Outline Introduction Theory and method Numerical examples
Simple Salt Model
Sigsbee Salt Model
Conclusions

17. Simple Salt Model 4 8 (a) True velocity model x (km) z (km) 8 x (km) 54 8
(a) True velocity model
x (km)
z (km) 8
x (km) 5
(b) CSG
t (s) 1 5
V(km/s) 4 8
(c) Initial Velocity Model
x (km)
z (km) 4 8
(d) RTM image
x (km)
z (km)

18. Angle-domain Crosscorrelation4 8
(a) Initial Velocity Model
x (km)
z (km)
(b) Angle-domain Crosscorrelation ∆𝝉 𝒛 𝜽
𝑓 𝑧,𝜃,∆𝜏
(c) Angle-domain Crosscorrelation
∆𝜏=𝛼 ( tan 𝜃 ) 2
∆𝜏: 𝛼: 𝜃:
time delay
curvature
reflection angle ∆𝝉 𝒛 𝜽
𝑓 𝑥,𝑧,𝜃,∆𝜏

19. Inversion Result (a) Initial velocity model z (km) 5 4 84 8
(a) Initial velocity model
x (km)
z (km) 1 5
Velocity(km/s) 4
(b) Inverted velocity model
z (km)
x (km) 8

20. Inversion Result 4 8 (a) RTM image x (km) z (km) 4 (b) RTM image4 8
(a) RTM image
x (km)
z (km) 4
(b) RTM image
z (km)
x (km) 8

21. Outline Introduction Theory and method Numerical examples
Simple Salt Model
Sigsbee Salt Model
Conclusions

22. Sigsbee Model (a) True velocity model (b) Initial velocity model z(km)6 12
z(km)
x(km) 6 12
z(km)
x(km)
Vinitial = 0.85 Vtrue 6 12
z(km)
x(km)
(c) RTM image
1.5
4.5
Velocity (km/s)

23. Initial Velocity Model6 12
z(km)
x(km) 6
-50°
+50°
CIG
Semblance
Crosscorrelation
-0.2
z(km)
z(km)
∆𝜏(𝑠)
∆𝜏=𝛼 ( tan 𝜃 ) 2 6
0.2 𝜃
-0.04 𝛼
0.04
-50° 𝜃
+50°

24. Initial Velocity Model6 12
z(km)
x(km)
Crosscorrelation 6
-50°
+50°
CIG
Semblance
-0.2
z(km)
z(km)
∆𝜏(𝑠)
∆𝜏=𝛼 ( tan 𝜃 ) 2
0.2 6 𝜃
-0.04 𝛼
0.04
-50° 𝜃
+50°

25. Initial Velocity Model6 12
z(km)
x(km) 6
-50°
+50°
CIG
Semblance
Crosscorrelation
-0.2
z(km)
z(km)
∆𝜏(𝑠)
∆𝜏=𝛼 ( tan 𝜃 ) 2
0.2 6 𝜃
-0.04 𝛼
0.04
-50° 𝜃
+50°

26. Inverted Velocity Model6 12
z(km)
x(km) 6
-50°
+50°
CIG
Semblance
Crosscorrelation
-0.2
z(km)
z(km)
∆𝜏(𝑠)
∆𝜏=𝛼 ( tan 𝜃 ) 2 6
0.2 𝜃
-0.04 𝛼
0.04
-50° 𝜃
+50°

27. Inverted Velocity Model6 12
z(km)
x(km)
Semblance 6
-50°
+50°
CIG
Crosscorrelation
-0.2
z(km)
z(km)
∆𝜏(𝑠)
∆𝜏=𝛼 ( tan 𝜃 ) 2 6
0.2 𝜃
-0.04 𝛼
0.04
-50° 𝜃
+50°

28. Inverted Velocity Model6 12
z(km)
x(km) 6
-50°
+50°
CIG
Semblance
Crosscorrelation
-0.2
z(km)
z(km)
∆𝜏(𝑠)
∆𝜏=𝛼 ( tan 𝜃 ) 2 6
0.2 𝜃
-0.04 𝛼
0.04
-50° 𝜃
+50°

29. RTM Image 6 12 z(km) x(km) (a) RTM image using initial velocity 6 126 12
z(km)
x(km)
(a) RTM image using initial velocity 6 12
z(km)
x(km)
(b) RTM image using inverted model

30. Outline
Introduction
Theory and method
Numerical examples
Conclusions

31. Velocity Inversion Methods
Data space
Image space
Ray-based tomography
Full Wavform inversion
Ray-based MVA
Wave-equ. MVA
Inversion
Wave-equ. traveltime
inversion
(Tomography)
Wave-equ. traveltime
inversion
The current velocity inversion methods can be divided into two types. One type defines the objective function in data space, such as traveltime tomography and full waveform inversion. The other type defines the objective function in model space, such as ray-based MVA and wave-equation-based MVA.
(MVA)

32. Angle-domain Wave-equation Reflection Traveltime Inversion
Traveltime inversion without high-frequency approximation
Misfit function somewhat linear with respect to velocity perturbation.
Wave-equation inversion less sensitive to amplitude
Multi-arrival traveltime inversion
Beam-based reflection traveltime inversion
To exploit the strengths and avoid the weaknesses of both traveltime tomography and full waveform inversion, wave-equation-based transimission traveltime inversion was developed to invert the velocity model. This method tried to invert the transmission traveltime informaiton using the method similar to full waveform inversion.

33. Thank you for your attention